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Volumn 42, Issue 3, 2004, Pages 1032-1061

A posteriori error estimates for discontinuous galerkin time-stepping method for optimal control problems governed by parabolic equations

Author keywords

A posteriori error analysis; Discontinuous galerkin method; Finite element approximation; Optimal control

Indexed keywords

APPROXIMATION THEORY; ERROR ANALYSIS; FINITE ELEMENT METHOD; MATHEMATICAL MODELS; OPTIMAL CONTROL SYSTEMS; SET THEORY; THEOREM PROVING;

EID: 21244485903     PISSN: 00361429     EISSN: None     Source Type: Journal    
DOI: 10.1137/S0036142902397090     Document Type: Article
Times cited : (67)

References (46)
  • 1
    • 0031102111 scopus 로고    scopus 로고
    • A posteriori error estimation in finite element analysis
    • M. AINSWORTH AND J. T. ODEN, A posteriori error estimation in finite element analysis, Comput. Methods Appl. Mech. Engrg., 142 (1997), pp. 1-88.
    • (1997) Comput. Methods Appl. Mech. Engrg. , vol.142 , pp. 1-88
    • Ainsworth, M.1    Oden, J.T.2
  • 2
    • 0030192433 scopus 로고    scopus 로고
    • Mesh adaption and optimization techniques in magnet design
    • P. ALOTTO, P. GIRDINIO, P. MOLFINO, AND M. NERVI, Mesh adaption and optimization techniques in magnet design, IEEE Trans. Magnetics, 32 (1996), pp. 2954-2957.
    • (1996) IEEE Trans. Magnetics , vol.32 , pp. 2954-2957
    • Alotto, P.1    Girdinio, P.2    Molfino, P.3    Nervi, M.4
  • 3
    • 0024705324 scopus 로고
    • Convergence of finite element approximation to state constrained convex parabolic boundary control problems
    • W. ALT AND U. MACKENROTH, Convergence of finite element approximation to state constrained convex parabolic boundary control problems, SIAM J. Control Optim., 27 (1989), pp. 718-736.
    • (1989) SIAM J. Control Optim. , vol.27 , pp. 718-736
    • Alt, W.1    Mackenroth, U.2
  • 5
    • 84966215690 scopus 로고
    • Some a posteriori error estimators for elliptic partial differential equations
    • R. E. BANK AND A. WEISER, Some a posteriori error estimators for elliptic partial differential equations, Math. Comp., 44 (1985), pp. 283-301.
    • (1985) Math. Comp. , vol.44 , pp. 283-301
    • Bank, R.E.1    Weiser, A.2
  • 7
    • 0034461033 scopus 로고    scopus 로고
    • Adaptive finite element methods for optimal control of partial differential equations: Basic concept
    • R. BECKER, H. KAPP, AND R. RANNACHER, Adaptive finite element methods for optimal control of partial differential equations: Basic concept, SIAM J. Control Optim., 39 (2000), pp. 113-132.
    • (2000) SIAM J. Control Optim. , vol.39 , pp. 113-132
    • Becker, R.1    Kapp, H.2    Rannacher, R.3
  • 9
    • 0034483495 scopus 로고    scopus 로고
    • Constants in Clement-interpolation error and residual based a posteriori estimates in finite element methods
    • C. CARSTENSEN AND S. A. FUNKEN, Constants in Clement-interpolation error and residual based a posteriori estimates in finite element methods, East-West J. Numer. Math., 8 (2000), pp. 153-175.
    • (2000) East-West J. Numer. Math. , vol.8 , pp. 153-175
    • Carstensen, C.1    Funken, S.A.2
  • 11
    • 0000076512 scopus 로고    scopus 로고
    • Runge-Kutta discontinuous Galerkin methods for convection-dominated problems
    • B. COCKBURN AND C. W. SHU, Runge-Kutta discontinuous Galerkin methods for convection-dominated problems, J. Sci. Comput., 16 (2001), pp. 173-261.
    • (2001) J. Sci. Comput. , vol.16 , pp. 173-261
    • Cockburn, B.1    Shu, C.W.2
  • 12
    • 0026106415 scopus 로고
    • Adaptive finite element methods for parabolic problems I: A linear model problem
    • K. ERIKSSON AND C. JOHNSON, Adaptive finite element methods for parabolic problems I: A linear model problem, SIAM J. Numer. Anal., 28 (1991), pp. 43-77.
    • (1991) SIAM J. Numer. Anal. , vol.28 , pp. 43-77
    • Eriksson, K.1    Johnson, C.2
  • 14
    • 0001188018 scopus 로고
    • Time discretization of parabolic problems by the discontinuous Galerkin method
    • K. ERIKSSON, C. JOHNSON, AND V. THOMEE, Time discretization of parabolic problems by the discontinuous Galerkin method, RAIRO Model. Math. Anal. Numer., 19 (1985), pp. 611-643.
    • (1985) RAIRO Model. Math. Anal. Numer. , vol.19 , pp. 611-643
    • Eriksson, K.1    Johnson, C.2    Thomee, V.3
  • 15
    • 0002491694 scopus 로고
    • Approximation of a class of optimal control problems with order of convergence estimates
    • F. S. FALK, Approximation of a class of optimal control problems with order of convergence estimates, J. Math. Anal. Appl., 44 (1973), pp. 28-47.
    • (1973) J. Math. Anal. Appl. , vol.44 , pp. 28-47
    • Falk, F.S.1
  • 16
    • 0008414510 scopus 로고
    • Approximation of an elliptic control problem by the finite element method
    • D. A. FRENCH AND J. T. KING, Approximation of an elliptic control problem by the finite element method, Numer. Funct. Anal. Appl., 12 (1991), pp. 299-315.
    • (1991) Numer. Funct. Anal. Appl. , vol.12 , pp. 299-315
    • French, D.A.1    King, J.T.2
  • 18
    • 0000238396 scopus 로고
    • On the approximation of the solution of an optimal control problem governed by an elliptic equation
    • T. GEVECI, On the approximation of the solution of an optimal control problem governed by an elliptic equation, RAIRO Anal. Numer., 13 (1979), pp. 313-328.
    • (1979) RAIRO Anal. Numer. , vol.13 , pp. 313-328
    • Geveci, T.1
  • 19
    • 0001149281 scopus 로고
    • Analysis and finite element approximation of optimal control problems for stationary Navier-Stokes equations with Dirichlet controls
    • M. D. GUNZBURGER, L. HOU, AND T. SVOBODNY, Analysis and finite element approximation of optimal control problems for stationary Navier-Stokes equations with Dirichlet controls, RAIRO Model. Math. Anal. Numer., 25 (1991), pp. 711-748.
    • (1991) RAIRO Model. Math. Anal. Numer. , vol.25 , pp. 711-748
    • Gunzburger, M.D.1    Hou, L.2    Svobodny, T.3
  • 20
    • 21844496580 scopus 로고
    • Analysis and finite element approximation of an optimal control problem in electrochemistry with current density controls
    • L. Hou AND J. C. TURNER, Analysis and finite element approximation of an optimal control problem in electrochemistry with current density controls, Numer. Math., 71 (1995), pp. 289-315.
    • (1995) Numer. Math. , vol.71 , pp. 289-315
    • Hou, L.1    Turner, J.C.2
  • 22
    • 0000402458 scopus 로고
    • Adaptive finite element methods for the obstacle problem
    • C. JOHNSON, Adaptive finite element methods for the obstacle problem, Math. Models Methods Appl. Sci., 2 (1992), pp. 483-487.
    • (1992) Math. Models Methods Appl. Sci. , vol.2 , pp. 483-487
    • Johnson, C.1
  • 23
    • 0001255117 scopus 로고
    • Numerics and hydrodynamic stability: Toward error control in computational fluid dynamics
    • C. JOHNSON, R. RANNACHER, AND M. BOMAN, Numerics and hydrodynamic stability: Toward error control in computational fluid dynamics, SIAM J. Numer. Anal., 32 (1995), pp. 1058-1079.
    • (1995) SIAM J. Numer. Anal. , vol.32 , pp. 1058-1079
    • Johnson, C.1    Rannacher, R.2    Boman, M.3
  • 24
    • 0020133102 scopus 로고
    • Finite element approximation of parabolic time optimal control problems
    • G. KNOWLES, Finite element approximation of parabolic time optimal control problems, SIAM J. Control Optim., 20 (1982), pp. 414-427.
    • (1982) SIAM J. Control Optim. , vol.20 , pp. 414-427
    • Knowles, G.1
  • 26
    • 0021431120 scopus 로고
    • Ritz-Galerkin approximation of the time optimal boundary control problem for parabolic systems with Dirichlet boundary conditions
    • I. LASIECKA, Ritz-Galerkin approximation of the time optimal boundary control problem for parabolic systems with Dirichlet boundary conditions, SIAM J. Control Optim., 22 (1984), pp. 477-499.
    • (1984) SIAM J. Control Optim. , vol.22 , pp. 477-499
    • Lasiecka, I.1
  • 27
    • 0141645572 scopus 로고    scopus 로고
    • Adaptive finite element approximation for distributed elliptic optimal control problems
    • R. LI, W. B. LIU, H. P. MA, AND T. TANG, Adaptive finite element approximation for distributed elliptic optimal control problems, SIAM J. Control Optim., 41 (2002), pp. 1321-1349.
    • (2002) SIAM J. Control Optim. , vol.41 , pp. 1321-1349
    • Li, R.1    Liu, W.B.2    Ma, H.P.3    Tang, T.4
  • 31
    • 0000769409 scopus 로고
    • Optimality conditions for strongly monotone variational inequalities
    • W. B. LIU AND J. E. RUBIO, Optimality conditions for strongly monotone variational inequalities, Appl. Math. Optim., 27 (1993), pp. 291-312.
    • (1993) Appl. Math. Optim. , vol.27 , pp. 291-312
    • Liu, W.B.1    Rubio, J.E.2
  • 32
    • 0035519074 scopus 로고    scopus 로고
    • Error estimates approximation of optimization problems governed by nonlinear operators
    • W. B. LIU AND D. TIBA, Error estimates approximation of optimization problems governed by nonlinear operators, Numer. Funct. Anal. Optim., 22 (2001), pp. 953-972.
    • (2001) Numer. Funct. Anal. Optim. , vol.22 , pp. 953-972
    • Liu, W.B.1    Tiba, D.2
  • 33
    • 0035734315 scopus 로고    scopus 로고
    • Quasi-norm local error estimators for p-Laplacian
    • W. B. LIU AND N. YAN, Quasi-norm local error estimators for p-Laplacian, SIAM J. Numer. Anal., 39 (2001), pp. 100-127.
    • (2001) SIAM J. Numer. Anal. , vol.39 , pp. 100-127
    • Liu, W.B.1    Yan, N.2
  • 34
    • 0035733415 scopus 로고    scopus 로고
    • A posteriori error analysis for convex boundary control problems
    • W. B. LIU AND N. YAN, A posteriori error analysis for convex boundary control problems, SIAM J. Numer. Anal., 39 (2001), pp. 73-99.
    • (2001) SIAM J. Numer. Anal. , vol.39 , pp. 73-99
    • Liu, W.B.1    Yan, N.2
  • 35
    • 0035566398 scopus 로고    scopus 로고
    • A posteriori error analysis for convex distributed optimal control problems
    • W. B. LIU AND N. YAN, A posteriori error analysis for convex distributed optimal control problems, Adv. Comput. Math., 15 (2001), pp. 285-309.
    • (2001) Adv. Comput. Math. , vol.15 , pp. 285-309
    • Liu, W.B.1    Yan, N.2
  • 36
    • 0242641435 scopus 로고    scopus 로고
    • A posteriori error estimates for optimal control problems governed by parabolic equations
    • W. B. LIU AND N. YAN, A posteriori error estimates for optimal control problems governed by parabolic equations, Numer. Math., 93 (2003), pp. 497-521.
    • (2003) Numer. Math. , vol.93 , pp. 497-521
    • Liu, W.B.1    Yan, N.2
  • 37
    • 0019899110 scopus 로고
    • Convergence of approximations vs. regularity of solutions for convex, control-constrained optimal-control problems
    • K. MALANOWSKI, Convergence of approximations vs. regularity of solutions for convex, control-constrained optimal-control problems, Appl. Math. Optim., 8 (1981), pp. 69-95.
    • (1981) Appl. Math. Optim. , vol.8 , pp. 69-95
    • Malanowski, K.1
  • 38
    • 0032049678 scopus 로고    scopus 로고
    • Adaptive topology optimization of elastoplastic structures
    • K. MAUTE, S. SCHWARZ, AND E. RAMM, Adaptive topology optimization of elastoplastic structures, Structural Optimization, 15 (1998), pp. 81-91.
    • (1998) Structural Optimization , vol.15 , pp. 81-91
    • Maute, K.1    Schwarz, S.2    Ramm, E.3
  • 39
    • 0001697550 scopus 로고
    • The Ritz-Galerkin procedure for parabolic control problems
    • R. S. MCKNIGHT AND W. E. BOSARGE, JR., The Ritz-Galerkin procedure for parabolic control problems, SIAM J. Control Optim., 11 (1973), pp. 510-524.
    • (1973) SIAM J. Control Optim. , vol.11 , pp. 510-524
    • Mcknight, R.S.1    Bosarge Jr., W.E.2
  • 44
    • 0000197665 scopus 로고    scopus 로고
    • Error estimates for the discretization of state constrained convex control problems
    • D. TIBA AND F. TROLTZSCH, Error estimates for the discretization of state constrained convex control problems, Numerical Funct. Anal. Optim., 17 (1996), pp. 1005-1028.
    • (1996) Numerical Funct. Anal. Optim. , vol.17 , pp. 1005-1028
    • Tiba, D.1    Troltzsch, F.2
  • 45
    • 0003009171 scopus 로고
    • Semidiscrete Ritz-Galerkin approximation of nonlinear parabolic boundary control problems-Strong convergence of optimal control
    • F. TROLTZSCH, Semidiscrete Ritz-Galerkin approximation of nonlinear parabolic boundary control problems-Strong convergence of optimal control, Appl. Math. Optim., 29 (1994), pp. 309-329.
    • (1994) Appl. Math. Optim. , vol.29 , pp. 309-329
    • Troltzsch, F.1


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